非线性发展方程的小模板简化Pade格式

在有理逼近的紧致格式的理论基础上，采用特别的统一的Pade逼近形式，构造了针对高阶非线性发展方程的、简单小模板的差商格式．不仅保持了格式的四阶精度，而且还可以采用追赶法求解得到的3对角矩阵，或者采用三阶Runge-Kuna法直接求解积分．计算效果通过多种算例表明是十分令人满意的．相对于其他差分格式，此方法具有模板较小而精度保持四阶的优点．     

高阶精度耗散加权紧致非线性格式

基于构造耗散紧致线性格式的方法,发展了新的高阶精度耗散加权紧致非线性格式(DWCNS). 通过Fourier分析方法讨论了DWCNS的耗散与色散特性. 从修正波数来看,DWCNS 在光滑区的精度与显式迎风偏斜的5阶精度格式接近. 发展了边界格式和靠近边界格式,分析了均匀网格和拉伸网格上的渐近稳定性,并讨论了在多维Euler方程和Navier-Stokes方程中的应用. 几个典型的无黏/有黏算例表明DWCNS对间断有很好的捕捉能力,对边界层的分辨精度也很高,并具有很好的收敛性（含有强激波流场计算,其均方根残差可以收敛到机器零）,尤其是在网格设计合理的情况下能抑制TVD和ENO格式均无法克服的涡量场的非物理波动.     

非线性发展方程的小模板简化Padé格式

在有理逼近的紧致格式的理论基础上,采用特别的统一的Pad啨逼近形式,构造了针对高阶非线性发展方程的、简单小模板的差商格式·　不仅保持了格式的四阶精度,而且还可以采用追赶法求解得到的3对角矩阵,或者采用三阶Runge_Kutta法直接求解积分·　计算效果通过多种算例表明是十分令人满意的·　相对于其他差分格式,此方法具有模板较小而精度保持四阶的优点·     

对论文"任意精度的三点紧致显格式及其在CFD中的应用"的评论

证明了林建国等(林建国,谢志华,周俊陶,任意精度的三点紧致显格式及其在CFD中的应用.应用数学和力学,2007,28(7):843-852)提出的紧致显格式与传统的差分格式实质相同,是传统差分格式的另一表达形式,并不具有紧致格式的优点.尽管如此,但这种表达形式更紧凑,推导获得高精度的差分表达式相对于传统的Taylor展开求待定系数的方法也更加简单.     

一类TVD型的迎风紧致差分格式

给出一种迎风型TVD(total variation diminishing)格式的构造方法,该方法通过限制器来抑制线性紧致格式在模拟间断流场时的非物理波动,可构造出非线性TVD型紧致格式(CTVD)．然后采用该法构造出了3阶和5阶的TVD型紧致格式,并通过模拟一维组合波和Riemann问题,二维激波-涡相互干扰和激波-边界层相互作用等来考察它们的性能．数值实验表明了该类格式的高阶精度和分辨率,且过间断基本无振荡．     

具有三阶精度的数值微分紧致格式及其应用

本文从微分代数精度概念出发 ,引入了数值微分的紧致性概念 ,并且构造了在一般意义下的具有三点三阶精度的数值微分格式 ,以及在等距节点这种特殊情况下的计算格式 .最后 ,通过数值实验证实了格式的精度 .     

关于“交错网格紧致差分格式和满足等价性的压力Poisson方程”一文的两点说明

“交错网格紧致差分格式和满足等价性的压力Poisson方程”（19：1（1997），83－90）作者注：1．四阶格式要求边条件至少三阶精度．而（2．13）只有二阶精度，要得到三阶，我们可以象（2．14）那样，在（2．13）左端加2．为了保证四阶Runge－Kutta方法（对非定常边条件）的精度，我们将在下一文中用如下中间层边条件，其中见（2．17）．##F56关于“交错网格紧致差分格式和满足等价性的压力Poisson方程”一文的两点说明@于欣$中国科学院力学研究所     

求解一维定常对流扩散反应方程的混合型紧致差分格式

借助显式紧致格式和隐式紧致格式的思想,基于截断误差余项修正,并结合原方程本身,构造出了一种求解一维定常对流扩散反应方程的高精度混合型紧致差分格式.格式仅用到三个点上的未知函数值及一阶导数值,而一阶导数值利用四阶Pade格式进行计算,格式整体具有四阶精度.数值实验结果验证了格式的精确性和可靠性.     

求解对流扩散方程的高阶紧致Padé有限体积格式

<正>自然界许多物理现象都可用对流扩散方程来描述,如质量、能量以及动量守恒问题等.实际应用问题中的对流扩散方程往往比较复杂,难以求出精确解,因此研究其数值求解方法具有十分重要的意义.对流扩散方程的经典解法对于解光滑问题可以得到较好的计算结果,但对于大梯度问题以及边界层等问题,会产生较大误差.紧致格式使用较少的模板可以获得较高的精度,因此高精度紧致方法成了近年来的研究热点[1-4].针对已有高阶紧致格式在分辨率和守恒性方面的问题,本文借鉴文献[4][5]中的思想构造了非定常对     

求解对流扩散方程的一致四阶紧致格式

目前,许多高精度差分格式,由于未成功地构造与其精度匹配的稳定的边界格式,不得不采用低精度的边界格式.本文针对对流扩散方程证明了存在一致四阶紧致格式,它的边界点的计算格式和内点的计算格式的截断误差主项保持一致,给出了具体内点和边界格式;并分析了此半离散格式的渐近稳定性.数值结果表明该格式是四阶精度;在对流占优情况下,本文边界格式的数值结果比四阶精度的显式差分格式的的数值结果的数值振荡小,取得了不错的效果,理论结果得到了数值验证;驱动方腔数值结果显示,本文对N-S方程的离散格式具有很好的可靠性,适合对复杂流体流动的数值模拟和研究.     

A high‐order ADI finite difference scheme for a 3D reaction‐diffusion equation with neumann boundary condition

In this article, we extend the fourth‐order compact boundary scheme in Liao et al. (Numer Methods Partial Differential Equations 18 (2002), 340–354) to a 3D problem and then combine it with the fourth‐order compact alternating direction implicit (ADI) method in Gu et al. (J Comput Appl Math 155 (2003), 1–17) to solve the 3D reaction‐diffusion equation with Neumann boundary condition. First, the reaction‐diffusion equation is solved with a compact fourth‐order finite difference method based on the Padé approximation, which is then combined with the ADI method and a fourth‐order compact scheme to approximate the Neumann boundary condition, to obtain fourth order accuracy in space. The accuracy in the temporal dimension is improved to fourth order by applying the Richardson extrapolation technique, although the unconditional stability of the numerical method is proved, and several numerical examples are presented to demonstrate the accuracy and efficiency of the proposed new algorithm. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Integrated fast and high‐accuracy computation of convection diffusion equations using multiscale multigrid method

We present an explicit sixth‐order compact finite difference scheme for fast high‐accuracy numerical solutions of the two‐dimensional convection diffusion equation with variable coefficients. The sixth‐order scheme is based on the well‐known fourth‐order compact (FOC) scheme, the Richardson extrapolation technique, and an operator interpolation scheme. For a particular implementation, we use multiscale multigrid method to compute the fourth‐order solutions on both the coarse grid and the fine grid. Then, an operator interpolation scheme combined with the Richardson extrapolation technique is used to compute a sixth‐order accurate fine grid solution. We compare the computed accuracy and the implementation cost of the new scheme with the standard nine‐point FOC scheme and Sun–Zhang's sixth‐order method. Two convection diffusion problems are solved numerically to validate our proposed sixth‐order scheme. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011     

An IMEX‐BDF2 compact scheme for pricing options under regime‐switching jump‐diffusion models

In this paper, an implicit‐explicit two‐step backward differentiation formula (IMEX‐BDF2) together with finite difference compact scheme is developed for the numerical pricing of European and American options whose asset price dynamics follow the regime‐switching jump‐diffusion process. It is shown that IMEX‐BDF2 method for solving this system of coupled partial integro‐differential equations is stable with the second‐order accuracy in time. On the basis of IMEX‐BDF2 time semi‐discrete method, we derive a fourth‐order compact (FOC) finite difference scheme for spatial discretization. Since the payoff function of the option at the strike price is not differentiable, the results show only second‐order accuracy in space. To remedy this, a local mesh refinement strategy is used near the strike price so that the accuracy achieves fourth order. Numerical results illustrate the effectiveness of the proposed method for European and American options under regime‐switching jump‐diffusion models.     

A high‐order finite difference discretization strategy based on extrapolation for convection diffusion equations

We propose a new high‐order finite difference discretization strategy, which is based on the Richardson extrapolation technique and an operator interpolation scheme, to solve convection diffusion equations. For a particular implementation, we solve a fine grid equation and a coarse grid equation by using a fourth‐order compact difference scheme. Then we combine the two approximate solutions and use the Richardson extrapolation to compute a sixth‐order accuracy coarse grid solution. A sixth‐order accuracy fine grid solution is obtained by interpolating the sixth‐order coarse grid solution using an operator interpolation scheme. Numerical results are presented to demonstrate the accuracy and efficacy of the proposed finite difference discretization strategy, compared to the sixth‐order combined compact difference (CCD) scheme, and the standard fourth‐order compact difference (FOC) scheme. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 20: 18–32, 2004.     

A new numerical scheme for the nonlinear Schrödinger equation with wave operator

In this paper, a new compact finite difference scheme is proposed for a periodic initial value problem of the nonlinear Schrödinger equation with wave operator. This is an explicit scheme of four levels with a discrete conservation law. The unconditional stability and convergence in maximum norm with order \(O(h^{4}+\tau ^{2})\) are verified by the energy method. Those theoretical results are proved by a numerical experiment and it is also verified that this scheme is better than the previous scheme via comparison.     

一维非线性Schrödinger 方程的两个无条件收敛的守恒紧致差分格式

本文对一维非线性 Schrödinger 方程给出两个紧致差分格式, 运用能量方法和两个新的分析技 巧证明格式关于离散质量和离散能量守恒, 而且在最大模意义下无条件收敛. 对非线性紧格式构造了 一个新的迭代算法, 证明了算法的收敛性, 并在此基础上给出一个新的线性化紧格式. 数值算例验证 了理论分析的正确性, 并通过外推进一步提高了数值解的精度.     

Symbolic computation of high order compact difference schemes for three dimensional linear elliptic partial differential equations with variable coefficients

We present a symbolic computation procedure for deriving various high order compact difference approximation schemes for certain three dimensional linear elliptic partial differential equations with variable coefficients. Based on the Maple software package, we approximate the leading terms in the truncation error of the Taylor series expansion of the governing equation and obtain a 19 point fourth order compact difference scheme for a general linear elliptic partial differential equation. A test problem is solved numerically to validate the derived fourth order compact difference scheme. This symbolic derivation method is simple and can be easily used to derive high order difference approximation schemes for other similar linear elliptic partial differential equations.     

Compact difference scheme for two-dimensional fourth-order nonlinear hyperbolic equation

High-order compact finite difference method for solving the two-dimensional fourth-order nonlinear hyperbolic equation is considered in this article. In order to design an implicit compact finite difference scheme, the fourth-order equation is written as a system of two second-order equations by introducing the second-order spatial derivative as a new variable. The second-order spatial derivatives are approximated by the compact finite difference operators to obtain a fourth-order convergence. As well as, the second-order time derivative is approximated by the central difference method. Then, existence and uniqueness of numerical solution is given. The stability and convergence of the compact finite difference scheme are proved by the energy method. Numerical results are provided to verify the accuracy and efficiency of this scheme.